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In this paper, we present translation hypersurfaces of semi-Euclidean spaces with zero scalar curvature. In addition, we prove that translation hypersurfaces with constant scalar curvature must have zero scalar curvature in the semi-Euclidean space $ \mathbb{R}_{q}^{n+1} $ for $ n\geq 3 $.

In this paper, homothetical and translation lightlike graphs, which are generalizations of homothetical and translation lightlike hypersurfaces are investigated in the semi-Euclidean space $ \mathbb{R}_{q}^{n+2} $, respectively. We prove that all homothetical and all translation lightlike graphs are locally the hyperplanes. According to this fact, both of these graphs are minimal.

In this paper, we present translation hypersurfaces of semi-Euclidean spaces with zero scalar curvature. In addition, we prove that translation hypersurfaces with constant scalar curvature must have zero scalar curvature in the semi-Euclidean space ... . [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for bi-null curves in R62 to be osculating, normal or rectifying curves in terms of their curvature functions. [ABSTRACT FROM AUTHOR]

In this paper, we establish a correlation between the bihyperbolic numbers set and the semi-Euclidean space. There are three different norms on the semi-Euclidean space that allow us to define three different hypersurfaces on semi-Euclidean space. Hence, we construct some topological structures on these hypersurfaces called norm e, s, and t topologies. On the other hand, we introduce hyperbolic e, s, and t topologies on the bihyperbolic numbers set. Moreover, by using the idempotent and spectral representations of the bihyperbolic numbers, we introduce new topologies on the bihyperbolic numbers set. [ABSTRACT FROM AUTHOR]

In the present paper, we give the notion of k-type bi-null Cartan slant helices in R 62, where k ∈ {1, 2, 3, 4, 5, 6}. We give the necessary and sufficient conditions for bi-null Cartan curves to be k-type slant helices in terms of their curvature functions. [ABSTRACT FROM AUTHOR]

In this paper, homothetical and translation lightlike graphs, which are generalizations of homothetical and translation lightlike hypersurfaces are investigated in the semi-Euclidean space Rn+2 q, respectively. We prove that all homothetical and all translation lightlike graphs are locally the hyperplanes. According to this fact, both of these graphs are minimal. [ABSTRACT FROM AUTHOR]

In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display them in a table that contains all information about Legendrian dualities. [ABSTRACT FROM AUTHOR]

In this paper, we establish a correlation between the bihyperbolic numbers set and the semi-Euclidean space. There are three different norms on the semi-Euclidean space that allow us to define three different hypersurfaces on semi-Euclidean space. Hence, we construct some topological structures on these hypersurfaces called norm e, s, and t topologies. On the other hand, we introduce hyperbolic e, s, and t topologies on the bihyperbolic numbers set. Moreover, by using the idempotent and spectral representations of the bihyperbolic numbers, we introduce new topologies on the bihyperbolic numbers set.

In the present paper, we introduce the notion of Cartan‐framed generalized null Mannheim curves in four‐dimensional semi‐Euclidean space with index 2, E24. We obtain the Cartan (or Frenet) frames and curvature functions of the generalized Mannheim mate curve by the help of curvature functions and Cartan frames of the generalized null Mannheim curve. We proved that there are no generalized null Mannheim curves with timelike principal normal N in E24. The related examples and their projected images are also given. [ABSTRACT FROM AUTHOR]

In this paper, kinematics of semi‐real quaternionic curve in semi‐Euclidean space E24 is obtained in terms of its curvature functions. The evolution equation of Frenet frame and curvatures of quaternionic curve are obtained. Also, examples of evolution equations of curvatures are given. [ABSTRACT FROM AUTHOR]

In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display them in a table that contains all information about Legendrian dualities.

In this paper, we prove that any non-constant real rational function appears as a time transformation of a caloric morphism, mapping which preserves caloric functions, between semi-eucledean spaces. [ABSTRACT FROM AUTHOR]

According to the Frenet equations of the null curves in semi-Euclidean 4-space, the existence conditions and the geometrical characterizations of the Bertrand curves of the null curves are given in this paper. The examples and the graphs of the Bertrand pairs with two different conditions are also given in order to supplement the conclusion of this paper more intuitively.

In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results.

Copyright of Adiyaman University Journal of Science & Technology / Adıyaman Üniversitesi Fen Bilimleri Dergisi is the property of Adiyaman University, Institute of Science / Adiyaman Universitesi Fen Bilimleri Enstitusu and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the present paper, we give the notion of k-type bi-null Cartan slant helices in R36, where k ∈ {1, 2, 3, 4, 5, 6}. We give the necessary and sufficient conditions for bi-null Cartan curves to be k-type slant helices in terms of their curvature functions. [ABSTRACT FROM AUTHOR]

In the present paper, we give the notion of k-type bi-null slant helices in $$\mathbb {R}_{2}^{5}$$ , where $$k\in \left\{ 0,1,2,3,4\right\} $$ . We give the necessary and sufficient conditions for bi-null curves to be k-type slant helices in terms of their curvature functions. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the geometric properties of two classes of ruled surfaces whose directrix curve is a bi-null curve in semi-Euclidean space $${\mathbb {R}}_{2}^{5}$$ . We get some classes of flat marginally trapped surfaces in $${\mathbb {R}}_{2}^{5}$$ . [ABSTRACT FROM AUTHOR]

In this paper, we study Lorentzian stationary surfaces related to bi-null curves in . Considering a Gauss-like map, we get Lorentzian stationary surfaces and marginally trapped surfaces in the pseudo-sphere . [ABSTRACT FROM AUTHOR]

In this paper, pseudo-spherical pseudo null and partially null curves are defined by using curvature functions in 424, respectively. Also, some new characterizations for pseudo null and partially null curves are obtained in 424, respectively. [ABSTRACT FROM AUTHOR]

Copyright of Journal of the Institute of Science & Technology / Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi is the property of Igdir University, Institute of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We show a formula for curvatures of curves in a semi-Euclidean space (or pseudo-sphere) with respect to Frenet-Serre type frame in terms of volumes. We also investigate versality of height unfolding and distance squared unfolding for a curve. [ABSTRACT FROM AUTHOR]

In this paper, we study generalized Cartan null Bertrand curves in semi-Euclidean 4-space $${\mathbb{E}_{2}^{4}}$$ with index 2. [ABSTRACT FROM AUTHOR]

In this paper, we study bi-null curves in semi-Euclidean 6-space with index 3, $${\mathbb{R}_{3}^{6}}$$ . We construct the Frenet frame and Cartan curvature functions of bi-null curves in $${\mathbb{R}_{3}^{6}}$$ . Also we discuss some properties of bi-null Cartan curves in terms of the Cartan curvatures. [ABSTRACT FROM AUTHOR]

In this paper, we study generalized pseudo null Bertrand curves in semi-Euclidean 4-space $$\mathbb {E}_{2}^{4}$$ with index 2. We get the necessary and sufficient conditions for pseudo null curves to be generalized Bertrand curves and we give some examples. [ABSTRACT FROM AUTHOR]

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space $${\mathbb{R}^{n+p}_{q}}$$ of index q, with $${1\leq q\leq p}$$ . Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of $${\mathbb{R}^{n+p}_{q}}$$ be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in $${\mathbb{R}^{n+p}_{p}}$$ and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz-Minkowski space $${\mathbb{R}^{n+1}_{1}}$$ . [ABSTRACT FROM AUTHOR]

In this work, we defined a quaternionic B2-slant helix in semi-Euclidean space E42. Then we gave Frenet formulae for the quaternionic curve in semi-Euclidean space E42 . Also, we investigated some necessary and sufficient conditions for a space curve to be a quaternionic B2-slant helix according to quaternionic curves in semi-Euclidean space E42 . [ABSTRACT FROM AUTHOR]

In this paper, we show that hyperquadrics are infinitesimally rigid in a semi-Euclidean space. We also show that hypersurfaces of hyperquadrics cut by hyperplanes not passing through the origin are infinitesimally rigid in the hyperquadrics, whereas those cut by hyperplanes through the origin are not infinitesimally rigid in hyperquadrics. Furthermore, we prove that any hypersurface in a semi-Euclidean space containing some open subset of a hyperplane is not infinitesimally rigid. [ABSTRACT FROM AUTHOR]

In this study, we define the osculating spheres of a semi real quaternionic curve in semi-Euclidean spaces 피1² and 피24. We give the equation of the osculating spheres with respect to Frenet frames {t0,n10,n20} and {T0,N10,N20,N30}. [ABSTRACT FROM AUTHOR]

We give the notion of bi-null Cartan curves in semi-Euclidean spaces of index 2, together with the unique Frenet frame and the Cartan curvatures. We also discuss some properties of bi-null Cartan curves in terms of the Cartan curvatures. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we studied quaternionic slant helices which we called semi-real quaternionic B 2 slant helix in four dimensional semi-Euclidean space . Also we define the harmonic curvature functions for semi-real quaternionic curves and we give a vector field D which we call Darboux quaternion for semi-real quaternionic B 2 slant helix. And then we obtained some characterizations for semi-real quaternionic B 2 slant helix in the terms of the harmonic curvatures and the Darboux vector field D. [Copyright &y& Elsevier]

We give the necessary and sufficient condition for a lightlike hypersurface in R42 with integrable screen distribution to be minimal. Using the condition we can get many minimal lightlike hypersurfaces in R42 which are not totally geodesic. [ABSTRACT FROM AUTHOR]

According to the Frenet equations of the null curves in semi-Euclidean 4-space, the existence conditions and the geometrical characterizations of the Bertrand curves of the null curves are given in this paper. The examples and the graphs of the Bertrand pairs with two different conditions are also given in order to supplement the conclusion of this paper more intuitively. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the Frenet equations of a pseudo null and a partially null curves, lying fully in the semi–Euclidean space $$ R^{4}_{2} $$ , and classify all such curves with constant curvatures. [ABSTRACT FROM AUTHOR]

In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results. [ABSTRACT FROM AUTHOR]

We prove a cylinder theorem for isometric pluriharmonic immersions of complete Kähler manifolds into semi-Euclidean spaces under an assumption concerning the index of relative nullity. [ABSTRACT FROM AUTHOR]